Diversification


BUSI 721, Fall 2022
JGSB, Rice University

Kerry Back

Why diversify your investments?

  • What is the risk of one coin flip for $10?
    • The mean outcome is zero.
    • The possible deviations from the mean are +10 and -10.
  • What is the risk of two coin flips for $5 each?
    • The mean outcome is zero.
    • The possible deviations are
      • +10 (prob 1/4),
      • -10 (prob 1/4) and
      • 0 (prob 1/2).

More baskets is better

Variance and standard deviation

  • Variance is expected squared deviation from mean
    • Flipping once for $10 \(\Rightarrow\) variance = 100
    • Flipping twice for $5 \(\Rightarrow\) variance = (1/4) \(\times\) 100 + (1/4) \(\times\) 100 = 50
  • Standard deviation is square root of variance
    • Flipping once for $10 \(\Rightarrow\) std dev = 10
    • Flipping twice for $5 \(\Rightarrow\) std dev = 7.07

Investment interpretation

  • Suppose we start with $100 before we flip
  • There are two independent assets, each with possible returns of \(\pm\) 10%
    • Flipping once for $10 \(\sim\) putting all $100 in a single asset
    • Flipping twice for $5 \(\sim\) putting $50 in each
  • Call the assets 1 and 2 with random returns \(r_1\) and \(r_2\)
  • The portfolio return \(r_p = (1/2) r_1 + (1/2) r_2\) is less risky than either \(r_1\) or \(r_2\).

Portfolio Returns

  • Consider a $100,000 portfolio with 40% in one asset (Asset 1) and 60% in a second (Asset 2).
  • Suppose asset 1 \(\uparrow\) 20%. $40,000 \(\rightarrow\) $48,000
  • Suppose asset 2 \(\uparrow\) 10%. $60,000 \(\rightarrow\) $66,000
  • $100,000 \(\rightarrow\) $114,000
  • So, up 14% = 0.4 \(\times\) 20% + 0.6 \(\times\) 10%
  • The portfolio return is \(r_p = w_1r_1 + w_2r_2\) where \(w_i\) is the fraction of the portfolio invested in asset \(i\).

Expected portfolio return

  • \(\bar{r}_1\) = expected return of asset 1
  • \(\bar{r}_2\) = expected return of asset 2
  • The expected portfolio return is

\[ \bar{r}_p = w_1 \bar{r}_1 + w_2 \bar{r}_2\]

Portfolio variance

  • The deviation of the portfolio return from its mean is

\[w_1r_1 + w_2r_2 - (w_1 \bar{r}_1 +w_2\bar{r}_2)\]

  • This equals

\[w_1(r_1 - \bar{r}_1) + w_2(r_2 - \bar{r}_2)\]

  • \((a+b)^2 = a^2 + b^2 + 2ab\), so the squared deviation is the sum of \(w_1^2(r_1-\bar{r}_1)^2\), \(w_2^2(r_2 - \bar{r}_2)^2\), and \(2w_1w_2(r_1-\bar{r}_1)(r_2-\bar{r}_2)\).

  • Expected value of \(w_1^2(r_1-\bar{r}_1)^2\) is \(w_1^2 \text{var}(r_1)\).
  • Expected value of \(w_2^2(r_2 - \bar{r}_2)^2\) is \(w_2^2 \text{var}(r_2)\).
  • Expected value of \(2w_1w_2(r_1-\bar{r}_1)(r_2-\bar{r}_2)\) is \(2w_1w_2\text{cov}(r_1,r_2)\).
  • Portfolio variance is

\[w_1^2 \text{var}(r_1) + w_2^2 \text{var}(r_2) + 2 w_1 w_2 \text{cov}(r_1, r_2)\]

2 Coin Flips Example

  • std dev \(r_1 = 10\text{%}\) \(\Rightarrow\) var \(r_1 = 100\text{%}^2\)
  • std dev \(r_2 = 10\text{%}\) \(\Rightarrow\) var \(r_2 = 100\text{%}^2\)
  • coin flips are independent, so cov \(= 0\)
  • with weights = 1/2, portfolio variance is

\[\left(\frac{1}{2}\right)^2 100\text{%}^2 + \left(\frac{1}{2}\right)^2 100\text{%}^2 = 50\text{%}^2\]

  • portfolio std dev is \(\sqrt{50}\text{%}= 7.07\text{%}\)

std dev of APPL is 0.01848158065635535

  • \(var(r_{p})\) is smaller when the covariance is smaller.
  • Covariance is correlation times product of standard deviations, so \(var(r_{p})\) is smaller when the covariance is smaller.

Portfolio Variance with n Assets

\[\small var(r_{p})=\sum_{i=1}^n w_{i}^2var(r_i)+2\sum_{i=1}^n \sum_{j=i+1}^n w_{i}w_{j}cov(r_{i},r_{j})\]

There are \(\frac{n(n-1)}{2}\) covariance terms.

A Simple Case

\(n\) assets, all variances are the same = \(\sigma^2\),
all correlations are the same = \(\rho\),
all weights = \(\frac{1}{n}\).

\[\small var(r_{p})=\sum_{i=1}^n (\frac{1}{n})^2\sigma^2+2\sum_{i=1}^n \sum_{j=i+1}^n (\frac{1}{n})^2\rho\sigma^2\]

\[\small var(r_{p})=\frac{1}{n}\sigma^2+\frac{n-1}{n}\rho\sigma^2 \rightarrow \rho\sigma^2\]

3C

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2A

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Portfolio Std Dev < Weighted Average of Asset Std Devs

\[\small (w_{A}\sigma_{A}+w_{B}\sigma_{B})^2=w_{A}^2\sigma_{A}^2+w_{B}^2\sigma_{B}^2+2w_{A}w_{B}\sigma_{A}\sigma_{B}\]

same as \(var_{p}\) except no correlation in the last term. So,

\[(w_{A}\sigma_{A}+w_{B}\sigma_{B})^2>var(r_{p})\]

Some Diversification Usually Lowers Risk

Compare holding only asset B to holding some mix of A and B.

Substitute \(w_{B}=(1-w_{A})\). Variance in general is

\[\small w_{A}^2\sigma_{A}^2+(1-w_{A})^2\sigma_{B}^2+2w_{A}(1-w_{A})\rho\sigma_{A}\sigma_{B}\]

Derivative with respect to \(w_{A}\) evaluated at \(w_{A}\)=0:

\[\small 2w_{A}\sigma^2_{A}-2(1-w_{A})\sigma^2_{B}+2(1-2w_{A})\rho\sigma_{A}\sigma_{B}=-2\sigma^2_{B}+2\rho\sigma_{A}\sigma_{B}\]

Risk is lowered by \(w_{A}\) \(\uparrow\) when \(\rho\sigma_{A}<\sigma_{B}\)